Another realization of the category of modules ove

arXiv:math/0010270v4[math.QA]8Apr2002ANOTHERREALIZATIONOFTHECATEGORYOFMODULESOVERTHESMALLQUANTUMGROUPSERGEYARKHIPOVANDDENNISGAITSGORYIntroduction0.1.Letgbeasemi-simpleLiealgebra.Givenarootofunity(cf.Sect.1.2),onecanconsidertworemarkablealgebras,Uℓanduℓ,calledthebigandthesmallquantumgroup,respectively.LetUℓ-modanduℓ-moddenotethecorrespondingcategoriesofmodules.Itisexplainedin[16]and[1]thattheformerisananalogincharacteristic0ofthecategoryofalgebraicrepresentationsofthecorrespondinggroupGoverafieldofpositivecharacteristic,andthelatterisananalogofthecategoryofrepresentationsofitsfirstFrobeniuskernel.Itisafactofcrucialimportance,thatalthoughUℓisintroducedasanalgebrade-finedbyanexplicitsetofgeneratorsandrelations,thecategoryUℓ-mod(or,rather,itsregularblock,cf.Sect.5.1)canbedescribedinpurelygeometricterms,asperversesheavesonthe(enhanced)affineflagvarietyeFl,cf.Sect.6.5.Thisisobtainedbycom-biningtheKazhdan-LusztigequivalencebetweenquantumgroupsandaffinealgebrasandtheKashiwara-TanisakilocalizationofmodulesovertheaffinealgebraoneFl.Thispaperisafirststepintheprojectoffindingageometricrealizationofthecategoryuℓ-mod.Weshouldsayrightawaythatonesuchrealizationalreadyexists,andisasubjectof[6].However,wewouldliketoinvestigateotherdirections.WeweremotivatedbyasetofconjecturesproposedbyB.Feigin,E.FrenkelandG.Lusztig,which,ontheonehand,tiethecategoryuℓ-modtothe(stillhypothetical)categoryofperversesheavesonthesemi-infiniteflagvariety(cf.[7],[8]),andontheotherhand,relatethelattertothecategoryofmodulesovertheaffinealgebraatthecriticallevel.Sincewealreadyknowthegeometricinterpretationformodulesoverthebigquantumgroup,itisanaturalideatofirstexpressuℓ-modentirelyintermsofUℓ-mod.Thisisexactlywhatwedointhispaper.0.2.Accordingto[13],thereisafunctorFr∗fromthecategoryoffinite-dimensionalrepresentationsoftheLanglandsdualgrouptoUℓ-mod.Inparticular,weobtainabi-functor:ˇG-mod×Uℓ-mod→Uℓ-mod:V,M→Fr∗(V)⊗M.WeintroducethecategoryC(AG,OˇG)tohaveasobjectsUℓ-modulesM,whichsatisfytheHeckeeigen-condition,inthesenseof[5].Inotherwords,anobjectofC(AG,OˇG)consistsofM∈Uℓ-modandacollectionofmapsαV:Fr∗(V)⊗M→V⊗M,whereVisthevectorspaceunderlyingtherepresen-tationV.ThemainresultofthispaperisTheorem2.4,whichstatesthatthereisanaturalequivalencebetweenC(AG,OˇG)anduℓ-mod.12SERGEYARKHIPOVANDDENNISGAITSGORYAsthereaderwillnotice,theproofofTheorem2.4isextremelysimple.However,itallowsonetogivethedesireddescriptionoftheregularblockuℓ-mod0ofthecategoryofuℓ-modulesintermsofperversesheavesontheenhancedaffineflagvarietysatisfyingtheHeckeeigen-condition,cf.Sect.6.4.Inafuturepublication,wewillexplainhowTheorem6.4canbeusedtodefineafunctorfromuℓ-mod0tothecategoryofperversesheavesonthesemi-infiniteflagvari-etyandtootherinterestingcategoriesthatariseinrepresentationtheory.Inparticular,uℓ-mod0obtainsaninterpretationintermsofthegeometricLanglandscorrespondence:itcanbethoughtofasacategoricalcounterpartofthespaceofIwahori-invariantvec-torsinasphericalrepresentation.Inanotherdirection,Theorem2.4hasasaconsequencethetheoremthatuℓ-modisequivalenttothecategoryofG[[t]]-integrablerepresentationsofthechiralHeckealgebra,introducedbyBeilinsonandDrinfeld.(Wedonotstatethistheoremexplicitly,becausethedefinitionofthechiralHeckealgebraisstillunavailabaleinthepublishedliterature.)0.3.Letusbrieflydescribethecontentsofthepaper.InSect.1werecallthebasicdefinitionsconcerningquantumgroups.InSect.2westateourmaintheoremanditsgeneralizationforpairsofbi-algebras(A,a).InSect.3weproveTheorem2.4inthegeneralsetting.InSect.4wediscussseveralcategoricalinterpretationsofTheorem2.4and,inparticular,itsvariantthatconcernsthegradedversion•uℓofuℓ.InSect.5wediscusstherelationbetweentheblockdecompositionsofUℓanduℓ.Finally,inSect.6weproveTheorem6.4,whichprovidesageometricinterpretationforthecategoryuℓ-mod0.Inthispaperweconsiderquantumgroupsatarootofunityofanevenorder,inordertobeabletoapplytheKazhdan-Lusztigequivalence.However,themainresulti.e.Theorem2.4holdsandcanbeprovedinexactlythesamewayinthecaseofarootofunityofanoddorder,withthedifferencethatinthedefinitionofthequantumFrobenius,theLanglandsdualgroupˇGmustbereplacedbyG.0.4.Acknowledgments.Themainideaofthispaper,i.e.Theorem2.4,occurredtousafteraseriesofconversationswithB.Feigin,M.FinkelbergandA.Braverman,towhomwewouldliketoexpressourgratitude.Inaddition,wewouldlikementionthatTheorem2.4,wasindependentlyandalmostsimultaneouslyobtainedbyB.FeiginandE.Frenkel.1.Quantumgroups1.1.Rootdata.LetGbeasemi-simplesimply-connectedgroup.LetTbetheCartangroupofGandlet(I,X,Y)bethecorrespondingrootdata,whereIisthesetofverticesoftheDynkindiagram,XisthesetofcharactersT−→Gm(i.e.theweightlatticeofG)andYisthesetofco-charactersGm−→T(i.e.thecorootlatticeofG).Wewilldenotebyh,ithecanonicalpairingY×X−→Z.Foreveryi∈I,αi∈X(resp.,MODULESOVERTHESMALLQUANTUMGROUP3ˇαi∈Y)willdenotethecorrespondingsimpleroot(resp.,coroot);fori,j∈Iwewilldenotebyai,jthecorrespondingentryoftheCartanmatrix,i.e.ai,j=hˇαi,αji.Let(·,·):X×X−→Qbethecanonicalinnerform.Inotherwords,||αi||2=2di,wheredi∈{1,2,3}istheminimalsetofintegerssuchthatthematrix(αi,αj):=di·ai,jiss